QCD Plasma Instabilities

and

Nonthermal Fixed Points

Jürgen Berges

Darmstadt University of Technology

Content

• Nonequilibrium instabilities (Weibel, Nielson-Olesen)

• Classical-statistical lattice gauge theory

• Characteristic time scales of SU(2), SU(3) pure gauge

• Instability-induced fermion production

• Kolmogorov wave turbulence vs. infrared fixed points

Nonequilibrium dynamics

Relativistic heavy-ion collisions explore

strong interaction matter starting from a

transient nonequilibrium state

Thermalization process?

Schematically:

• Characteristic nonequilibrium time scales? Relaxation? Instabilities?

tiso » O(1/gT)

Isotropization time tiso? In the absence of nonequilibrium instabilities:

tiso » O(1/g4T)

• Weibel instability:

Weibel ’59; … Mrowczynski ’88, ’93, ’94; Arnold, Lenaghan, Moore ‘03;

Romatschke, Strickland ‘03; very many since then…

oblate anisotropy ! Txx » Tyy À Tzz

• Nielsen-Olesen instability:

tiso » O(1/g1/2B1/2)

Nielsen, Olesen ’78; Chang, Weiss ’79; … Iwasaki ’08; Fujii, Itakura ’08 …

…

Short-time dynamics

• Anisotropy of the stress tensor Tij in a local rest frame:

“homogeneous“

background field

characteristic momentum of typical excitation

B z

z (force)

x (current)

y (magnetic field)

h{

,

}i

Nonperturbative description that contains all possible mechanisms:

Real-time lattice QCD

• Quantum simulation not possible so far (stochastic quantization?)

• Classical-statistical simulation: includes all fluctuations with

(includes, in particular, HTL / Vlasov)

physics of nonequilibrium instabilities ! high occupation numbers

• Classical simulations well tested for quantum evolutions in scalar theories:

p

Berges, Rothkopf, Schmidt,

PRL 101 (2008) 041603

(N=4)-component 4

parametric/spinodal

instability

i.e. “occupation numbers“ À 1h{ A(x) , A(y) }i À h[ A(x) , A(y) ]i

Classical-statistical lattice gauge field simulations

with À z (extreme anisotropy)

Normalized Gaussian probability functional

Here: = 0 / = s = 4, axial-temporal/Coulomb gauge

Wilson action:

Romatschke, Venugopalan (CGC); Berges, Gelfand, Scheffler, Sexty

Initial conditions:

P[A(t=0),tA(t=0)]: h A(t) A(t‘) i = s DA(0) DtA(0) P[A(0),tA(0)] A(t) A(t‘)

C is adjusted to obtain a given energy density , e.g. RHIC ~ 5-25 GeV/fm3

kT

/1/4 = 1

primary

secondary growth rates

Inverse primary growth rates:

Characteristic time scales

Berges, Scheffler, Sexty, PRD 77 (2008) 034504

e.g. RHIC ~ 5-25 GeV/fm3, LHC ~ 2 x RHIC

fast slow

fast: SU(2)

(Coulomb gauge)

Comparison SU(2) vs. SU(3)

) reduced primary growth rates by about 25% for given with /1/4 = 1

Berges, Gelfand, Scheffler, Sexty, PLB 677 (2009) 210

SU(3)

) Measured in units of the characteristic screening masses mT,SU(2) and

mT,SU(3), respectively, the primary growth rate is independent of Nc

Comparison SU(2) vs. SU(3)

) SU(3) / SU(2) = (3/4) £ ( SU(2) / SU(3) ) :

SU(2) = SU(3) i.e. 25% reduction, SU(2) / SU(3) = (8/3)1/4 i.e. only ~4%

C » 2

/1/4 = 1

• Initial conditions with faster isotropization/thermalization?

/ 1/4 ¿ 1 corresponds to rather “homogeneous“ field configurations (z / 1/4 ¿ 1)

pT

) growth rate increases

for smaller / 1/4

! Nielsen-Olesen?

SU(2)

Pressure

Fast bottom-up isotropization for momenta:

fast slow

pz . 1 GeV

Spatial Fourier transform of stress tensor T(t,x): PL(t,p) for ==3, PT(t,p)

/1/4 = 1

SU(2)

Kolmogorov wave turbulence

Berges, Scheffler, Sexty, arXiv:0811.4293 [hep-ph]

See however: Arnold, Moore PRD 73 (2006) 025006; Mueller, Shoshi, Wong, NPB 760 (2007) 145

• Different infrared behavior? Nonthermal IR fixed point?

(Infrared occupation number » 1/g2 ) strongly correlated)

• Scaling exponent close to the perturbative value = 4/3

t M

Kolmogorov

IR fixed point

Berges, Rothkopf, Schmidt, PRL 101 (2008) 041603

» k-2

» k-4 ?

Arnold, Moore `06

(Vlasov)

= 2 excluded!?

Berges, Scheffler, Sexty `08

(classical-statistical lattice QCD)

pe

rtu

rba

tive

(Ko

lmo

go

rov)

• Apparent discrepancy:

• Compare scalar instability dynamics (parametric resonance):

(perturbative)(nonperturbative)

not fully

understood

perturbatively

RG: ‘microscope‘ with varying resolution of length scale

~ 1/k

Fixed point: physics looks the same for ‘all‘ resolutions (in rescaled units)

scaling form, e.g. ‘length rescaled‘ correlator

Slow dynamics: Nonthermal RG fixed points

similarly, retarded propagator:

anomalous dimension

Typically not for all resolutions: • IR fixed point for k ! 0

• UV fixed point for k ! ‘1‘

‘occupation number‘ exponent

• vacuum:

• thermal:

• nonequilibrium:

Fluctuation-dissipation relation:

No fluctuation-dissipation relation:

spatial dimension d

dynamical exponent z

A) Hierarchy of infrared fixed point solutions for scalar N-component QFT:

Berges, Hoffmeister, NPB 813 (2009) 383

, ... (NLO 1/N)

Relativistic ( z ' 1) scalar inflaton in d = 3, ' 0 ) ' 4 , …

B) Ultraviolet fixed point solutions (classical-statistical limit, Kolmogorov):

Relativistic scalar inflaton in d = 3 ) ' 3/2 , 4/3 , …

No Kolmogorov turbulence in the far UV due to quantum corrections

Instability-induced fermion production

Quantum evolution of SU(2)L£SU(2)R linear sigma model (2PI 1/N to NLO):

• Fermion production proceeds with the

maximum primary boson growth rate!

Berges, Pruschke, Rothkopf, PRD 80 (2009) 023522

thermal equilibrium:

• Fast approach to Fermi-Dirac distribution

in the infrared!

Bosons

(spinodal)Fermions

V

g2 / = 1

fast slow

• Fast fermion growth induced via boson-fermion loop:

Boson

Fermion

(no IR scaling ! Pauli principle) • Scaling behavior at higher momenta

• Bosons practically unaffected for Yukawa couplings . O(1)

! Instability-induced fermion production can lead to substantial

deviations from standard production processes

1/s

» O(1)

Quantitatively: Classical-statistical lattice QCD with (quantum)

fermions can be simulated with well established techniques!

Conclusions

• Plasma instabilities for CGC type initial conditions ( / 1/4 = 1)

• ‘Bottom-up’ isotropization for p . 1 GeV,

i.e. (optimistically) about the range where hydro ‘works’

• (Perturbative) Kolmogorov wave turbulence with ' 4/3

Nonthermal IR fixed point in QCD?

• Initial conditions with faster isotropization/thermalization?

• Instability-induced fermion production can lead to substantial

deviations from standard production processes

! fast thermalization of low-momentum fermions on time scale

of maximum boson growth rate seen in linear sigma model

! » O(QCD) ?

-1 ' 1 – 2 fm/c